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At present one can not find a single counterexample to even a simplestversion of Frobenius primality test. The assessment of probability of themistake, presented in [I.B. Damgard and G.S.Frandsen, Journal of Cryptology,2006] is strongly overestimated. In the present paper, the properties of simpledivisors of FPP-numbers are proved. The lower bound for FPP are given.

Counterexamples for Frobenius primality test